How to deal with evolutionary/genetic fitness function that can have both negative and positive values?

Question

I am optimising function that can have both positive and negative values in pretty much unknown ranges, might be -100, 30, 0.001, or 4000, or -0.4 and I wonder how I can transform these results so I can use it as a fitness function in evolutionary algorithms and it can be optimised in a way that, for example, it can go from negative to positive along the optimisation process (first generation best chromosome can have -4.3 and best at 1000 generation would have 5.9). Although the main goal would always be to maximise the function.

Adding a constant value like 100 and then treating it simply as positive is not possible because like I said, the function might optimise different ranges of results in different runs for example (-10000 to +400 and in another run from -0.002 to -0.5).

Is there a way to solve this?

Answer 1

If I understand correctly your problem, you always want to maximize some function $h(x)$, which is defined as follows $h: \Gamma \rightarrow \mathbb{R}$, where $\Gamma$ is the space of genotypes. However, at every generation $g$, you don't know exactly $h(i)$ of each individual $i$, i.e., maybe in one generation $g$, all $h(i)$, for $i=1, \dots, N$ (where $N$ is the size of the population), are negative, but in the next generation $g+1$, due to the mutations and crossovers, that may not be the case anymore, so you don't know how to shift $h(i), \forall i$, so that they are all positive and you can compute the probability of being selected, assuming you are using the fitness proportionate selection.

If my intepretation is correct, then, at every generation $g$, you just need to find $w = \operatorname{min}_i h(i)$, then shift all $h(i)$ by adding $|w| + \epsilon$ (where $\epsilon$ can be zero or a very small number) to all $h(i)$, so you would compute the fitness as follows $f(i) = h(i) + |w| + \epsilon$, for all $i$. Here are all the steps to compute the probability of individual $i$ being selected $p(i)$

1. $w = \operatorname{min}_i h(i)$
2. $f(i) \leftarrow h(i) + |w| + \epsilon, \forall i$
3. $p(i) \leftarrow \frac{f(i)}{\sum_{j=1}^{N} f(j)}, \forall i$

This technique is known as windowing, which I have used in the past to solve the exact same problem that you seem to be trying to solve (check it here).